Electron means for generating trigonometric functions



Aug. 7, 1962 R. T. BYERLY 3,048,337

ELECTRON MEANS FOR GENERATING TRIGONOMETRIC FUNCTIONS Filed July 2, 19574 Sheets-Sheet 1 Fig. 3.

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Aug. 7, 1 962 R. T. BYERLY 3,048,337

ELECTRON MEANS FOR GENERATING TRIGONOMETRIC FUNCTIONS Filed July 2, 19574 Sheets-Sheet 2 Fig.6.

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Aug. 7, 1962 R T. BYERLY 3,048,337

ELECTRON MEANS FOR GENERATING TRIGONOMETRIC FUNCTIONS 4 Sheets-Sheet 5Filed July 2, 1957 Aug. 7, 1962 R. T. BYERLY 3,048,337

ELECTRON MEANS FOR GENERATING TRIGONOMETRIC FUNCTIONS Filed July 2, 19574 Sheets-Sheet 4 Unite This invention relates to an electronic means forgenerating trigonometric functions and more particularly to a circuitarrangement of high gain amplifiers capable of producing an output, amagnitude of which varies as the trigonometric function of a controlinput.

Some previous methods for developing trigonometri functions of dependentvariables involved the positioning of a rotating shaft for varying thepositions of a variable ta-p along a potentiometer, the resistance ofwhich is constructed to have a trigonometric function such as sine orcosine. The rotating shaft is usually operated by servo-mechanisms ofcomputing type. This method of generating trigonometric functions is offairly good static quality but falls short of the dynamic desirabilitiesdue to the physical limitations of the positioning system. The actualinertia of the mechanical components limits the speed of response to anundesirable slow rate. In addition, the potentiometer being generally awire wound type of device provides a stepped output rather than a smoothcontinuous output from minimum to maximum limits.

It is therefore an object of this invention to provide an electroniccircuit for generating trigonometric functions.

It is another object of this invention to provide an electronic circuitcap-able of developing trigonometric functions that are free from minutevoltage steps in varying from maximum to minimum limits or vice versa.

It is another object of this invention to provide an electronic circuitfor generating trigonometric functions that is free from output delaythat is caused by mechanical inertia.

It is another object of this invention to provide an electronic circuitfor generating trigonometric functions that is capable of providinginfinite output voltage values between the maximum and minimum voltagelimitations.

Other objects, purposes and characteristic features will become apparentas a description of the invention progresses.

In practicing this invention, there is provided high gain amplifiershaving voltage limit means for regulating the output of the amplifiercircuit to maximum values of the voltage limit means above and belowzero. The amplifier for the circuit is provided with a plurality ofinputs, one of which acts as a reference voltage and which is variableas a trigonometric function rate such as the sine. The two inputs, whencombined at the amplifier, cause the output voltage of the amplifier tobe alternately positive and negative for periods established by acomparison of the two input voltages. The output voltage of theamplifiers is then filtered and appears as a direct voltage, theamplitude of which varies as the trigonometric function of one of theinput voltages.

FIGURE 1 is a view of a circuit capable of generating a trigonometricfunction.

FIG. 2 is a view of circuit similar to FIG. 1 that is capable ofdeveloping trigonometric functions;

FIG. 3 is a curve representing typical input voltages for the circuitsof FIGS. 1 and 2;

'FIG. 4 is a curve representing the output voltages of the amplifiers ofFIGS. 1 and 2 prior to being filtered;

FIG. 5 is a curve representing the output voltages that could be foundat the output terminals of filters of FIGS.

1 and 2;

ice

FIG. 6 is a curve of still different magnitude voltages typical of thosecapable of being the input to the circuits of FIGS. 1 and 2;

FIG. 7 is a curve representing an output voltage of the amplifier priorto filtering of FIG. 2;

FIG. 8 is a curve of the output voltage of the filter of FIG. 2resulting from the input voltages of FIG. 6;

FIG. 9 is a view of a circuit capable of developing the sine function ofan input control voltage;

FIG. 10 is a view of a circuit capable of developing the sine functionof an input voltage over a greater range than the circuit of FIG. 9;

FIG. 11 is a view of a circuit capable of developing the cosine functionof an input voltage to the circuit;

FIG. l2 is a graphical representation showing the points at which outputswitching will occur in the circuit of FIG. 1 when the input controlvoltage is positive in polarity; and

FIG. 13 is a graphical representation showing the points at which outputswitching will occur for the circuit of FIG. 1 when the input controlvoltage is negative in polarity.

In each of the several views, similar parts bear like referencecharacters.

The circuit shown in FIG. 1 is a circuit capable of developing an outputvoltage, the magnitude of which is variable as a function of an inputreference voltage and an input control voltage. The input referencevoltage is preferably of a pure sine wave form which might berepresented as E sin wt, where E represents the maximum peak voltage ofthe sine wave reference voltage. This voltage will be designated as eThe control voltage is basically a direct current voltage. However, itmay vary in amplitude but at a rate far less than the rate of variationof the reference voltage 2 The direct current control voltage is for thepurpose of simplification designated e The range of e must be such thatE e E The circuit of FIG. 1 comprises an amplifier which is preferablyof high gain characteristics and which may have a gain as high asone-hundred million. The exact construction of FIG. 1 is not shown inthis invention since any suitable high gain type of amplifier can beused.

Amplifier 1 is provided with an input circuit 2 connected to the inputvoltage sources e and e through the resistors 3 and 4 respectively. Theresistors 3 and 4 generally are of approximately equal value, howeverunder certain circumstances, involving the use of this circuit theseresistors may be of unequal values as is necessary.

Amplifier 1 is provided with an output circuit 5 connected to the inputcircuit 6 of a suitable filter 7, the exact structure of which is notshown since it is not essential to this invention.

In order to provide both positive and negative voltage limits foramplifier 1, a pair of feedback or limiting circuits 8 and 9 connectingthe output circuit 5 of amplifier 1 to the input circuit 2 of amplifier1 is provided. The feedback or limiting circuit 8 comprises a battery 10and rectifier 11 connected in series, with the negative terminal of thebattery 10 being connected to the input circuit 2 of the amplifier 1.Likewise, the feedback or limiter circuit 9 comprises a battery 12 andrectifier 13 connected in series, with the positive terminal of thebattery connected to the input circuit 2 of amplifier 1.

With the application of voltages e and 6 to the input circuit 2 ofamplifier 1, the amplifier will attempt to provide an outputproportional to the combined input. When the output voltage, whether itbe positive or negative, reaches the value of the voltage of one of thefeedback or limiting circuits 8 or 9, the feedback circuit then becomesa feedback path which limits the output voltage to the level of thebattery Voltage found in the corresponding feedback path. For example,if we assume that the combined voltages e and e provides a positiveinput to the amplifier 1, the output voltage will rise negatively untilthe voltage is equal to the voltage of the DC. source or battery 12,since this is the only feedback path through which current due to theoutput voltage can pass, due to the rectifier 13 allowing passage andthe rectifier 11 blocking passage of this polarity voltage. When theoutput voltage equals the voltage of the battery 12, the battery thenbecomes a low resistance path between the output circuit of theamplifier 1 and the input circuit 2, thus causing the amplifier outputvoltage to be limited to the value of the feedback battery voltage 12.

It should be pointed out, that in order for the feedback voltage to bein a proper phase relationship to limit the output voltage to thevoltages E of the battery voltages and 12, it is necessary for theamplifier 1 to also phase shift its output 180. Without the phase shift,the feedback voltage merely tends to cause unstable characteristics inthe circuit shown.

The curve shown in FIG. 4 represents a typical output curve of thevoltage e and e due to the input voltages e and e of FIG. 3 beingapplied to the circuits of FIGS. 1 and 2. It can be seen that if wecombine the input voltages e and e and apply them to the amplifier 1 ofFIG. 1 that as long as the combined voltages result in a value greaterthan zero and with a phase reversal taking place in the amplifier 1, theoutput voltage of e will maintain a -E value of battery 12 until thereference voltage 0 when combined with the control voltage e arrives ata total input voltage to the amplifier of slightly less than zero. Atthis time, as the combined input voltages e and 2 start to go negative,the amplifier 1 produces a positive output voltage of E value. As longas the combined voltages e and e produce a negative input voltage to theamplifier 1, the output voltages e will maintain the E value. However,as soon as the reference voltage 2 starts to rise and rises to a pointcapable of producing a resultant positive input voltage to the amplifier1, the output 3 of amplifier 1 very rapidly moves from the positive Evalue to a negative E value. This operation continues as long as the twoinputs e and e are applied to the amplifier 1.

It should be pointed out that although the wave form of 0 shown in FIG.4 is shown as a square wave, it is true that the vertical lines betweenthe +E and E values actually follow lines that are proportional to thecurve of the negative voltage 2 Since the amplifier 1 is a high gaindevice, the E values are generally very small in comparison to the inputtimes the gain of the amplifier and thus for all practical purposes thetime period taken for the amplifier to reverse its output from +E to Eor vice versa is of extremely small duration in comparison to the timethat the voltage is maintained at the E levels.

The square wave voltage e is then passed into the filter 7' and resultsin an output voltage from the filter 7 of a value e represented by theoutput voltage a of FIG. 5. The magnitude of a is established by thealgebraic sum of the positive and negative portions of the wave formshown in FIG. 4. For example, the area of the square wave found abovethe zero line of FIG. 4 is substantially less than the area of areafound below the zero line thus resulting in a negative e value forFIG. 1. The circuit of FIG. 1 is capable of producing an e voltage equalto With the foregoing description of the device of FIGURE 1 and itsoperation, the relationship between e and e can be obtained by analysis.It is to be stated that the voltage 2 represents the sine of an angle 0which is assumed to lie in the first or fourth quadrant, the scale ofthe voltage being such that where 0 is measured in radians. This scalerelationship is indicated to be negative as a convenience to compensatefor phase reversal in the amplifier of FIGURE 1. Thus, positive valuesof 2 correspond to fourth quadrant values of 0 and negative values of ecorrespond to first quadrant values of 0 Consider now only values of 0in the first quadrant for which e will be negative. The correspondingswitching diagram for one cycle of the sinusoid is shown in FIGURE 12.The value of e will be e =+E from 0 to 1 6 'EB from L11 IO (12 e +E from0: to 21r Thus the value of e will be Note that d1=1l"-Xz Thensubstituting for 11 Now consider the angle 0 to lie in the fourthquadrant, specifying 0 numerically as with e positive. The switchingdiagram is as shown in FIGURE 13. The reference for angular measurementhas been shifted to avoid a more complicated notation. The values of eare given as Substituting for 0a;

As before, since switching occurs at a point '-B1 =E sill 01 Thus, thecircuit of FIGURE 1 is established as being an arcsin device operatingin the first and fourth quadrants. In general, if e +e O, then e wouldbe equal to E;;, if e +e 0, then e would be equalto +E This relationshipholds true as long as there is a phase reversal in amplifier 1 and anassociated feedback path.

The circuit of FIG. 2 is similar to that of FIG. 1 with the exceptionthat the limiting voltage sources E are now placed in the output circuitof the amplifier 1 in such a manner as to act as an output limitingdevice and not as a feedback limiter. With this arrangement, it isunnecessary to provide a phase reversal in amplifier 1. As pointed outpreviously, the phase reversal in the amplifier of FIG. 1 was necessaryto provide the proper phase relationship between the input voltage andthe feedback voltage. The limiting circuits 8 and 9 in FIG. 2 providehigh resistance for any output voltage on output circuit of amplifier 1until the battery voltage E of the proper limiting path 8 or 9 isexceeded. At this time, the limiting path becomes a low resistance pathto ground for any attempted increase in voltage above the value of thebattery voltage E If we again take a look at FIGS. 3, 4 and 5 whileconsidering the circuit arrangement of FIG. 2, we can see that FIG. 3discloses the input voltages to the balancing resistors 3 and 4 of FIG.2 in a manner identical with that of FIG. 1. The combined input voltagesare then applied to the input circuit 2 for the amplifier 1 thensuitably amplified and applied to the output circuit 5 as an amplifiedsignal. Since the amplifier 1 is again a high gain type of amplifier,the output voltage e can be assumedto have risen positively to a valueequal to the voltage +E of the proper limiting path. Since no phasereversal has taken place in amplifier 1, the output voltage e willappear at the maximum level of +E with the rectifier 11 and batterysource acting as a shunt path of low resistance for any voltage abovethe battery 10 voltage +E In a manner similar to the circuit of FIG. '1,the output voltage e remains at the +E value until the reference voltagee goes sufiiciently negative to make the resultant combined inputvoltages appear as a zero or slightly negative input to the amplifier 1.At this time, the output of the amplifier 1 moves rapidly to the -Evalue to remain at this level through the action of the current limitingpath 9 until the reference voltage e again rises to a point capable ofallowing a slightly positive input voltage to amplifier 1.

The output voltage e is as previously pointed out, the average value ofthe +E and -E values described by the output voltage curve e This valueis shown in FIG. 5.

In order to show how the output voltage a can vary as a function of thetwo input voltages 2 and e the curves of FIGS. 6, 7 and 8 are shown. Ifwe again assume that the reference voltage e is again applied to theinput circuit 2 of the amplifier 1, and that the control voltage 2 offar less value is applied to the input circuit 2 of the amplifier 1, theoutput voltage e again being limited to maximum values of -I-E and --EWould be represented by the voltage curve e shown in FIG. 7. It ispointed out, however, that the length of time that the output voltage ofthe amplifier 1 remains positive is much shorter, and the length of timethat the voltage remains negative is much longer, than the voltage curveshown in FIG. 4. For this reason, when the output voltage 632 is passedthrough the filter 7, a filter output voltage e is greatly reduced as afunction of the input control voltage e It should be clear that aninfinite number of values for e can be obtained with a circuit of thisnature without the introduction of stepped voltages.

The circuit shown by FIG. 9 is one employing a closed loop feedbackcircuit of high gain degenerative nature. This circuit is capable ofoperation over a range merely from i 2 In this circuit the incomingcontrol voltage e is applied to an amplifier 14 provided with aresistance feedback circuit 15 including the resistor 16. The value ofthe resistor 16 determines the amount of gain in the operation of theamplifier =14. This gain may be set at any value desired or foundnecessary for the particular system in which this circuit may be used.For the sake of clarity, we will assume that the resistor 16 allows ahigh gain within the amplifier 14. The input control voltage e istherefore applied to the input of the amplifier 14 in combination with afeedback voltage, the value of which will be explained hereinafter. Ifwe assume that the input voltage to amplifier 14 is applied and resultsin an output voltage, this voltage is applied through -a resistor 17 tothe input of an amplifier 1 8. Amplifier 18 is identical with theamplifier 14 in that the amplifier is provided with a suitable feedbackcircuit 19 provided with a resistor 20. The feedback circuit 20 alsoestablishes the output of amplifier 18 in a manner similar to thatdescribed in connection with amplifier 14.

Amplifiers 14 and 18 when used with a feedback or limiting circuit mustalso provide for a phase reversal in order to allow the feedback voltageto be in the proper phase relationship to act as an establishingvoltage. It is pointed out that although the feedback circuit is shownas using a resistor, the circuit could utilize other impedance devicessuch as capacitors.

It should also be pointed out that an amplifier not having a feedbackcircuit (such as shown in FIG. 2) can be used. However, the feedbackcircuit is preferable due to the increased stability in the amplifieroutput. The use of feedback controlled amplifiers dictates the use of aseries of two such amplifiers since it is necessary to cause two phasereversals between the input and the output of the system for properphase output.

The output of amplifier 18 results in an output voltage 2 which isapplied through a feedback rmistor 3 to a feedback circuit that isidentical in configuration and function with that described inconnection with FIG. 1. Since this circuit is identical with FIG. 1, itis also necessary to provide an input reference voltage e in a mannersimilar to FIG. 1. The output voltage of this feedback circuit thereforecan be designated as e since it is identical in character with theoutput voltage previously described in connection with FIG. 1. Since theoutput voltage 2 of the feedback circuit is a trigonometric function, aspreviously described, of the control voltage e applied to feedbackamplifier 1, through the feedback circuit 21 and since a phase reversaltakes place in the feedback amplifier 1, the voltage e applied throughthe resistor 22 acts as a degenerative voltage to the input voltage eThis closed loop circuit then becomes stable when the feedback voltage ematches the input control voltage e; as modified by the resistors 3 and22 before appliciation to the amplifier 14. Under normal conditions, theresistors 3 and 22 would probably be of similar resistance value,however, depending upon the circuit application in a system, theresistance values may be varied to suit the particular application.

In FIGURE 9, the voltage e is the same in nature as the voltage e inFIGURE 1, that is, it is proportional to the sine of an angle in thefirst or fourth quadrant. Since e is an appropriate sinusoid, 2 is anangle 0 in the first or fourth quadrant. But the nature of thedegenerative feedback causes e and c of FIGURE 9 to be in very closeagreement. Therefore, e may be interpreted as an 7 angle and e as thesine of that angle, the angle being retricted to the range -'n'/ 2 to+1r/2.

The modification shown in FIG. is similar to the circuit shown in FIG.9, with the exception that the input circuit to the amplifier 14 isprovided with circuitry capable of allowing operation overa greaterrange. The range in the case of the circuit shown in FIG. 10 can berepresented as 31r 31:- T1 0, T radians In other words, the range ofthis circuit is three times that of the circuit shown in FIG. 9. Thegreater range can best he explained by taking an input control voltageof different levels assuming that the input control voltage starts atzero. In the range from zero to the value plus or minus E established bythe batteries 23 and 24 depending upon the polarity of the applied 2voltage, the voltage is applied directly through the conductor 25 andresistor 26 to the input of amplifier 14. The input voltage is alsoapplied through the resistors 23a and 24a to the rectifiers 23b and 24bone of which allows passage of the input voltage to the battery 23 or 24depending upon the input voltage polarity. Since the input voltagemagnitude is less than E battery voltage this input path is blocked. Asthe value of 2 is raised, however, to a point above the E level, aninput appears on the amplifier 27 to be amplified and applied to theamplifier 14 through the re sistor 28. In this case, the amplifier 27 isprovided with a feedback loop 29 provided with a resistor 39 capable ofadjusting the gain in the amplifier 27. The value of this resistor isestablished to provide a dsirable gain of 2. This amplification of theincoming signal is then applied in parallel with the incoming signalthrough the resistors 28 and 26 respectively to the input of theamplifier 14. The remainder of the circuit of FIG. 10 is identical withthat of FIG. 9 and the respective parts bear identical referencenumerals. It should be pointed out however, that the feedback voltage eapplied through the resistor 22 to the input of the amplifier 14 againadjusts the output voltage 2 to a value established when the feedackvoltage matches the input control voltages received from the resistors26 and 28.

The circuit of FIG. 11 is used to generate the cosine of an inputvoltage rather than the sine as previously shown in the previousmodifications. In this circuit, the basic control circuit starting withamplifier 14 is identical with that described in connection with FIGS. 9and 10. However, the input control voltage e is applied to the input ofamplifier 14 through additional circuitry capable of effectivelymultiplying the positive input control voltage e by -1. For example, ifwe assume that the input control voltage e is a positive voltage, it canbe seen that rectifier 31 would allow conduction of the input voltagethrough the resistor 32 to the input of an amplifier 33 capable of phasereversal as well as amplification. The amplifier 33 is provided withfeedback loop path 34 including a gain resistor 35 adjusted to a gainfigure in this case necessarily a gain of one. Since a phase reversaltakes place in the amplifier 33, the output voltage of the amplifier isnegative and applied through the matching resistor 36 to the inputcircuit of the amplifier 14.

If the input control voltage e had been of negative polarity, thenconduction would occur through the rectifier 37 allowing an input to theamplifier 38 also capable of phase reversal action. The amplifier 38 isalso provided with a feedback loop 39 provided with a gain resistor 40again adjusted to a gain of one. The amplifier output voltage of theamplifier 38 is a positive output voltage, due to the phase reversal inamplifier 38, which is then fed through a limiting resistor 41 to theinput terminal of the amplifier 33. Amplifier 33 again amplifies thesignal and provides another phase reversal and applies it to theamplifier 14. Since the control voltage e always results in a negativevoltage before being applied to the input of the amplifier 14, it isnecessary to provide a second reference voltage to the input ofamplifier 14. This input reference voltage is a positive voltage and inthis instance preferably of E value. For this reason when the appliedvoltage e is zero the output voltage 6 is at a maximum positive value.As the applied voltage e rises positively or negatively the outputvoltage e reduces as a cosine function of the applied voltage.

As previously pointed out in connection with FIGS. 9 and 10, the inputvoltage to the amplifier 14 is amplified by the amplifiers 14 and 18 andappears as voltage e; which in turn is fed back as an input into thefeedback circuit through the amplifier 1 to modify the input controlvoltage in a degenerative manner to establish the desired output levelof the cosine voltage a; of the input control voltage e It is againemphasized that in order for the output voltage of the amplifier 33 ofFIG. 11 to be the same whether the input voltage e is positive ornegative the gain of amplifiers 33 and 38 must be one.

Since numerous changes may be made in the above described constructionand different embodiments of the invention may be made without departingfrom the spirit and scope thereof, it'is intended that all the mattercontained in the foregoing description or shown in the accompanyingdrawings shall be interpreted as illustrative and not in a limitingsense.

I claim as my invention:

1. An electronic circuit for generating trigonometric functionscomprising amplifier means having an input circuit and an outputcircuit; means for connecting a control voltage to said input circuit;and feedback means connecting said output circuit to said amplifiermeans to limit any output of said amplifier means to a value establishedby said feedback means; said feedback means comprising a feedbackamplifier operably connected to amplify the output from said outputcircuit and reverse the phase thereof, means for connecting a varyingreference voltage to said feedback amplifier, and filter means forconnecting the output from said feedback amplifier to said amplifiermeans; said input circuit including circuit means for feeding thecontrol voltage directly to said amplifier means, and input amplifiermeans in parallel circuit relationship with said circuit means foramplifying the control voltage twice upon said control voltage exceedinga predetermined value and reversing the phase of said amplified controlvoltage to oppose the application of the control voltage to saidamplifier means.

2. An electronic circuit for generating trigonometric functionscomprising amplifier means having an input circuit and an outputcircuit; means for connecting a control voltage to said input circuit;and feedback means, including means for connecting a varying referencevoltage to said feedback means, connecting said output circuit to saidamplifier means to limit any output from said amplifier means to a valueestablished by said feedback means; said input circuit including circuitmeans for feeding the control voltage directly to said amplifier means,and input amplifier means connected in parallel circuit relationship tosaid circuit means, said input amplifier means comprising an inputamplifier having a gainof two for amplifying the control voltage,polarized voltage means connected in series circuit relationship withsaid input amplifier for blocking the application of the control voltageto said input amplifier until said control voltage exceeds apredetermined value; said input amplifier including phase shifting meansfor reversing the phase of said amplified control voltage to oppose theapplication of the control voltage to said amplifier means.

3.An electronic circuit for generating trigonometric functionscomprising amplifier means having an input circuit and an outputcircuit; means for connecting a control voltage to said input circuit;and feedback means,

including means for connecting a varying reference voltage of the form Esin wt to said feedback means, connecting said output circuit to saidamplifier means to limit any output from said amplifier means to a valueestablished by said feedback means; said input circuit including circuitmeans for feeding the control voltage directly to said amplifier means,and input amplifier means connected in parallel circuit relationship tosaid circuit means, said input amplifier means comprising an inputamplifier having a gain of two for amplifying the control voltage,polarized voltage means connected in series circuit relationship withsaid input amplifier for blocking the application of the control voltageto said input amplifier until said control voltage exceeds apredetermined value; said input amplifier including phase shifting meansfor reversing the phase of said amplified 10 control voltage to opposethe application of the control voltage to said amplifier means.

References Cited in the file of this patent UNITED STATES PATENTS2,652,194 Hirsch Sept. 15, 1953 2,710,348 Baum et a1. June 7, 19552,748,278 Smith May 29, 1956 2,849,181 Lehmann Aug. 26, 1958 OTHERREFERENCES A Palimpsest on the Electronic Analog Art (Paynter), 1955,page 107.

Diode Limiters Simulate Mechanical Phenomena by Merrill and Baum, pp.122 to 126, Electronics. November 1952.

